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 Course

Mathematics

Bachelor Mathematik (STO/PO 2024

0084e_k120

  • Analysis I

    0084eA1.1
    • 19202801 Lecture
      Analysis I (Elena Mäder-Baumdicker)
      Schedule: Di 10:00-12:00, Do 10:00-12:00 (Class starts on: 2025-10-14)
      Location: T9/Gr. Hörsaal (Takustr. 9)

      Comments

      Content:
      This is the first part of a three semester introduction into the basic mathematical field of Analysis. Differential and integral calculus in a real variable will be covered. Topics:

      1. fundamentals, elementary logic, ordered pairs, relations, functions, domain and range of a function, inverse functions (injectivity, surjectivity)
      2. numbers, induction, calculations in R, C
      3. arrangement of R, maximum and minimum, supremum and infimum of real sets, supremum / infimum completeness of R, absolute value of a real number, Q is dense in R
      4. sequences and series, limits, cauchy sequences, convergence criteria, series and basic principles of convergence
      5. topological aspects of R, open, closed, and compact real sets
      6. sequences of functions, series of functions, power series
      7. properties of functions, boundedness, monotony, convexity
      8. continuity, limits and continuity of functions, uniform continuity, intermediate value theorems, continuity and compactness
      9. differentiability, concept of the derivative, differentiation rules, mean value theorem, local and global extrema, curvature, monotony, convexity
      10. elementary functions, rational functions, root functions, exponential functions, angular functions, hyperbolic functions, real logarithm, inverse trigonometric functions, curve sketching
      11. beginnings of integral calculus

       

      Suggested reading

      Literature:

      • Bröcker, Theodor: Analysis 1, Spektrum der Wissenschaft-Verlag.
      • Forster, Otto: Analysis 1, Vieweg-Verlag.
      • Spivak, Michael: Calculus, 4th Edition.

      Viele Analysis Bücher sind auch über die Fachbibliothek der FU Berlin elektronisch verfügbar.

      Bei Schwierigkeiten mit den Grundbegriffen Menge, Abbildung etc. ist die folgende Ausarbeitung empfehlenswert:
    • 19202802 Practice seminar
      Tutorial: Analysis I (Elena Mäder-Baumdicker)
      Schedule: Mi 12:00-14:00, Mi 14:00-16:00, Fr 08:00-10:00, Fr 10:00-12:00 (Class starts on: 2025-10-15)
      Location: Mi A3/SR 119 (Arnimallee 3-5), Fr A3/SR 119 (Arnimallee 3-5), Fr A6/SR 025/026 Seminarraum (Arnimallee 6)

  • Lineare Algebra I

    0084eA1.4
    • 19201401 Lecture
      Linear Algebra I (Georg Loho)
      Schedule: Mo 08:00-10:00, Mi 08:00-10:00 (Class starts on: 2025-10-15)
      Location: A3/Hs 001 Hörsaal (Arnimallee 3-5)

      Comments

      Content:

      • Basic terms/concepts: sets, maps, equivalence relations, groups, rings,
      • fields
      • Linear equation systems: solvability criteria, Gauss algorithm
      • Vector spaces: linear independence, generating systems and bases, dimension,
      • subspaces, quotient spaces, cross products in R3
      • Linear maps: image and rank, relationship to matrices, behaviour under
      • change of basis
      • Dual vector spaces: multilinear forms, alternating and symmetric bilinear
      • forms, relationship to matices, change of basis
      • Determinants: Cramer's rule, Eigenvalues and Eigenvectors


      Prerequisites:

      Participation in the preparatory course (Brückenkurs) is highly recommended.

       

      Suggested reading

      • Siegfried Bosch, Lineare Algebra, 4. Auflage, Springer-Verlag, 2008;
      • Gerd Fischer, Lernbuch Lineare Algebra und Analytische Geometrie, Springer-Verlag, 2017;
      • Bartel Leendert van der Waerden, Algebra Volume I, 9th Edition, Springer 1993;

      Zu den Grundlagen

      • Kevin Houston, Wie man mathematisch denkt: Eine Einführung in die mathematische Arbeitstechnik für Studienanfänger, Spektrum Akademischer Verlag, 2012
    • 19201402 Practice seminar
      Practice seminar for Linear Algebra I (Georg Loho, Jan-Hendrik de Wiljes)
      Schedule: Mo 10:00-12:00, Mo 16:00-18:00, Mi 10:00-12:00, Do 08:00-10:00, Do 12:00-14:00, Fr 10:00-12:00 (Class starts on: 2025-10-15)
      Location: Mo A3/SR 115 (Arnimallee 3-5), Mo A6/SR 009 Seminarraum (Arnimallee 6), Mi A6/SR 009 Seminarraum (Arnimallee 6), Do A6/SR 007/008 Seminarraum (Arnimallee 6), Do A6/SR 032 Seminarraum (Arnimallee 6), Fr A6/SR 031 Seminarraum (Arnimallee 6)

  • Introductory Module: Numerical Mathematics II

    0280cA1.11
    • 19202101 Lecture
      Basic Module: Numeric II (Robert Gruhlke)
      Schedule: Mo 12:00-14:00, Mi 12:00-14:00 (Class starts on: 2025-10-15)
      Location: A3/Hs 001 Hörsaal (Arnimallee 3-5)

      Comments

      Description: Extending basic knowledge on odes from Numerik I, we first concentrate on one-step methods for stiff and differential-algebraic systems and then discuss Hamiltonian systems. In the second part of the lecture we consider the iterative solution of large linear systems.

      Target Audience: Students of Bachelor and Master courses in Mathematics and of BMS

      Prerequisites: Basics of calculus (Analysis I, II) linear algebra (Lineare Algebra I, II) and numerical analysis (Numerik I)
    • 19202102 Practice seminar
      Practice seminar for Basic Module: Numeric II (André-Alexander Zepernick)
      Schedule: Mi 10:00-12:00, Fr 08:00-10:00 (Class starts on: 2025-10-15)
      Location: A6/SR 025/026 Seminarraum (Arnimallee 6)

  • Introductory Module: Probability and Statistics II

    0280cA1.15
    • 19212901 Lecture
      Stochastics II (Felix Höfling)
      Schedule: Di 12:00-14:00, Do 08:00-10:00 (Class starts on: 2025-10-14)
      Location: A3/Hs 001 Hörsaal (Arnimallee 3-5)

      Additional information / Pre-requisites

      Prerequisite: Stochastics I  and  Analysis I — III.

      Comments

      Contents:

      • Basics: conditional expectation, characteristic function, convergence types, uniform integrability;
      • Construction of stochastic processes and examples: Gaussian processes, Lévy processes, Brownian motion;
      • martingales in discrete time: convergence, stopping theorems, inequalities;
      • Markov chains in discrete and continuous time: recurrence and transience, invariant measures.

      Suggested reading

      • Klenke: Wahrscheinlichkeitstheorie
      • Durrett: Probability. Theory and Examples.

      Weitere Literatur wird im Lauf der Vorlesung bekannt gegeben.
      Further literature will be given during the lecture.
    • 19212902 Practice seminar
      Practice seminar for Stochastics II (Felix Höfling)
      Schedule: Di 14:00-16:00 (Class starts on: 2025-10-14)
      Location: A6/SR 031 Seminarraum (Arnimallee 6)

      Comments

      Inhalt

       

       

      • This course is the sequel of the course of Stochastics I. The main objective is to go beyond the first principles in probability theory by introducing the general language of measure theory, and the application of this framework in a wide variety of probabilistic scenarios.
        More precisely, the course will cover the following aspects of probability theory:
      • Measure theory and the Lebesgue integral
      • Convergence of random variables and 0-1 laws
      • Generating functions: branching processes and characteristic functions
      • Markov chains
      • Introduction to martingales

       

       

  • Introductory Module: Differential Geometry I

    0280cA1.3
    • 19202601 Lecture
      Differential Geometry I (Konrad Polthier)
      Schedule: Mo 10:00-12:00, Mi 10:00-12:00 (Class starts on: 2025-10-15)
      Location: KöLu24-26/SR 006 Neuro/Mathe (Königin-Luise-Str. 24 / 26)

      Additional information / Pre-requisites

      For further information, see Lecture Homepage.

      Comments

      Topics of the lecture will be:

      • curves and surfaces in Euclidean space,
      • metrics and (Riemannian) manifolds,
      • surface tension, notions of curvature,
      • vector fields, tensors, covariant derivative
      • geodesic curves, exponential map,
      • Gauß-Bonnet theorem, topology,
      • connection to discrete differential geometry.

      This course is a BMS course and will be held in English on request.

      Prerequisits:

      Analysis I, II, III and Linear Algebra I, II

       

       

      Suggested reading

      Literature

      • W. Kühnel: Differentialgeometrie:Kurven - Flächen - Mannigfaltigkeiten, Springer, 2012
      • M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall
      • J.-H. Eschenburg, J. Jost: Differentialgeometrie und Minimalflächen, Springer, 2014
      • C. Bär: Elementare Differentialgeometrie, de Gruyter, 2001
    • 19202602 Practice seminar
      Practice seminar for Differential Geometry I (Tillmann Kleiner, Konrad Polthier)
      Schedule: Mi 08:00-10:00 (Class starts on: 2025-10-15)
      Location: A6/SR 031 Seminarraum (Arnimallee 6)

  • Introductory Module: Discrete Geometry I

    0280cA1.5
    • 19202001 Lecture
      Discrete Geometrie I (Christian Haase)
      Schedule: Di 10:00-12:00, Mi 12:00-14:00 (Class starts on: 2025-10-14)
      Location: A3/SR 120 (Arnimallee 3-5)

      Additional information / Pre-requisites

      Solid background in linear algebra. Knowledge in combinatorics and geometry is advantageous.

      Comments

      This is the first in a series of three courses on discrete geometry. The aim of the course is a skillful handling of discrete geometric structures including analysis and proof techniques. The material will be a selection of the following topics:
      Basic structures in discrete geometry

      • polyhedra and polyhedral complexes
      • configurations of points, hyperplanes, subspaces
      • Subdivisions and triangulations (including Delaunay and Voronoi)
      • Polytope theory
      • Representations and the theorem of Minkowski-Weyl
      • polarity, simple/simplicial polytopes, shellability
      • shellability, face lattices, f-vectors, Euler- and Dehn-Sommerville
      • graphs, diameters, Hirsch (ex-)conjecture
      • Geometry of linear programming
      • linear programs, simplex algorithm, LP-duality
      • Combinatorial geometry / Geometric combinatorics
      • Arrangements of points and lines, Sylvester-Gallai, Erdos-Szekeres
      • Arrangements, zonotopes, zonotopal tilings, oriented matroids
      • Examples, examples, examples
      • regular polytopes, centrally symmetric polytopes
      • extremal polytopes, cyclic/neighborly polytopes, stacked polytopes
      • combinatorial optimization and 0/1-polytopes

       

      For students with an interest in discrete mathematics and geometry, this is the starting point to specialize in discrete geometry. The topics addressed in the course supplement and deepen the understanding for discrete-geometric structures appearing in differential geometry, topology, combinatorics, and algebraic geometry.

       

       

       

       

       

       

       

       

       

      Suggested reading

      • G.M. Ziegler "Lectures in Polytopes"
      • J. Matousek "Lectures on Discrete Geometry"
      • Further literature will be announced in class.
    • 19202002 Practice seminar
      Practice seminar for Discrete Geometrie I (Sofia Garzón Mora, Christian Haase)
      Schedule: Mi 14:00-16:00 (Class starts on: 2025-10-15)
      Location: A6/SR 031 Seminarraum (Arnimallee 6)

  • Modules w/o courses in this semester

    20 Modules
    • Analysis II 0084eA1.2
    • Analysis III 0084eA1.3
    • Linear Algebra II 0084eA1.5
    • Computer-Oriented Mathematics I 0084eA1.6
    • Computer-Oriented Mathematics II 0084eA1.7
    • Numerik I 0084eA1.8
    • Stochastik I 0084eA1.9
    • Wissenschaftliches Arbeiten in der Mathematik 0084eB1.1
    • Algebra und Zahlentheorie 0084eB2.1
    • Functional Analysis 0084eB2.2
    • Complex Analysis 0084eB2.3
    • Geometry 0084eB2.4
    • Special Topics in Mathematics 0084eB2.5
    • Spezialthemen der reinen Mathematik 0084eB2.6
    • Spezialthemen der angewandten Mathematik 0084eB2.7
    • Visualisierung 0084eB2.8
    • Introductory Module: Algebra I 0280cA1.1
    • Introductory Module: Partial Differential Equations I 0280cA1.13
    • Introductory Module: Topology I 0280cA1.17
    • Introductory Module: Discrete Mathematics I 0280cA1.7